Advertisements
Advertisements
प्रश्न
Evaluate the following:
`tan 1/2(cos^-1 sqrt5/3)`
उत्तर
Let, `cos^-1 sqrt5/3=theta`
`=> costheta=sqrt5/3`
`=>2cos^2 theta/2-1=sqrt5/3`
`=>cos^2 theta/2=(3+sqrt5)/6`
`=>theta/2=cos^-1(sqrt((3+sqrt5)/6))`
`=tan^-1((sqrt(1-(sqrt((3+sqrt5)/6))^2))/(sqrt((3+sqrt5)/6)))`
`=tan^-1(sqrt(1-(3+sqrt5)/6)/sqrt(3+sqrt5/6))`
`=tan^-1((sqrt((3-sqrt5)/6))/(sqrt((3+sqrt5)/6)))`
`=tan^-1(sqrt((3-sqrt5)/(3+sqrt5)))`
`=tan^-1(sqrt(((3-sqrt5)(3-sqrt5))/((3+sqrt5)(3-sqrt5))))`
`=tan^-1(sqrt((3-sqrt5)^2/(9-5)))`
`=tan^-1((3-sqrt5)/2)`
i. e. , `1/2(cos^-1 sqrt5/3)=tan^-1 ((3-sqrt5)/2)`
`=>tan 1/2(cos^-1 sqrt5/3)=tan[tan^-1((3-sqrt5)/2)]`
`thereforetan 1/2(cos^-1 sqrt5/3)=(3-sqrt5)/2`
APPEARS IN
संबंधित प्रश्न
Show that:
`2 sin^-1 (3/5)-tan^-1 (17/31)=pi/4`
If `(sin^-1x)^2 + (sin^-1y)^2+(sin^-1z)^2=3/4pi^2,` find the value of x2 + y2 + z2
Find the principal values of the following:
`cos^-1(tan (3pi)/4)`
`sin^-1(sin pi/6)`
Evaluate the following:
`cos^-1(cos4)`
Evaluate the following:
`tan^-1(tan (7pi)/6)`
Evaluate the following:
`tan^-1(tan4)`
Evaluate the following:
`tan^-1(tan12)`
Evaluate the following:
`sec^-1(sec pi/3)`
Evaluate the following:
`sec^-1(sec (2pi)/3)`
Evaluate the following:
`sec^-1{sec (-(7pi)/3)}`
Evaluate the following:
`cot^-1{cot ((21pi)/4)}`
Write the following in the simplest form:
`tan^-1sqrt((a-x)/(a+x)),-a<x<a`
Write the following in the simplest form:
`sin^-1{(x+sqrt(1-x^2))/sqrt2},-1<x<1`
Evaluate the following:
`sin(sec^-1 17/8)`
Prove the following result
`sin(cos^-1 3/5+sin^-1 5/13)=63/65`
Evaluate:
`cos{sin^-1(-7/25)}`
`sin^-1x=pi/6+cos^-1x`
Solve the following:
`sin^-1x+sin^-1 2x=pi/3`
`tan^-1 1/4+tan^-1 2/9=1/2cos^-1 3/2=1/2sin^-1(4/5)`
`2tan^-1(1/2)+tan^-1(1/7)=tan^-1(31/17)`
Show that `2tan^-1x+sin^-1 (2x)/(1+x^2)` is constant for x ≥ 1, find that constant.
Write the value of tan−1x + tan−1 `(1/x)`for x > 0.
Write the value of cos−1 (cos 1540°).
Write the value of sin−1
\[\left( \sin( -{600}°) \right)\].
Write the value of cos\[\left( 2 \sin^{- 1} \frac{1}{3} \right)\]
Evaluate: \[\sin^{- 1} \left( \sin\frac{3\pi}{5} \right)\]
What is the principal value of `sin^-1(-sqrt3/2)?`
Write the principal value of `sin^-1(-1/2)`
Write the principal value of \[\cos^{- 1} \left( \cos680^\circ \right)\]
Write the value of \[\cos\left( \sin^{- 1} x + \cos^{- 1} x \right), \left| x \right| \leq 1\]
Write the value of \[\tan^{- 1} \left( \frac{1}{x} \right)\] for x < 0 in terms of `cot^-1x`
The positive integral solution of the equation
\[\tan^{- 1} x + \cos^{- 1} \frac{y}{\sqrt{1 + y^2}} = \sin^{- 1} \frac{3}{\sqrt{10}}\text{ is }\]
If x < 0, y < 0 such that xy = 1, then tan−1 x + tan−1 y equals
If 4 cos−1 x + sin−1 x = π, then the value of x is
If \[\sin^{- 1} \left( \frac{2a}{1 - a^2} \right) + \cos^{- 1} \left( \frac{1 - a^2}{1 + a^2} \right) = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right),\text{ where }a, x \in \left( 0, 1 \right)\] , then, the value of x is
If y = sin (sin x), prove that \[\frac{d^2 y}{d x^2} + \tan x \frac{dy}{dx} + y \cos^2 x = 0 .\]
tanx is periodic with period ____________.
